Auxiliary Separation Lemma

Disjoint convex sets are separable if one has nonempty core and the sets are disjoint.

Auxiliary Separation Lemma

Let $X$ be a real vector space and let $\Omega_1,\Omega_2\subset X$ be nonempty convex sets . Assume that core(Ω₁) $\neq\emptyset$ and $\Omega_1\cap\Omega_2=\emptyset$ .

Lemma: The sets $\Omega_1$ and $\Omega_2$ can be separated by a hyperplane .

Context: The proof reduces separation of two sets to separation of a point from a convex set by applying point-vs-set separation to the Minkowski difference $\Omega_1-\Omega_2$ .